Group #16778028 — subgroups

Subgroups — #16778028

Order 3328.

166 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718214	C₂	1	1	yes	maximal, characteristic
2	#13131187	C₂	1	1	yes	maximal, characteristic
3	D₁₃×#368	C₂	1	1	yes	maximal, characteristic
4	#11719092	—	2	1	no	—
5	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
6	D₁₃×#108	—	2	1	no	—
7	C₂×#88036	—	2	1	no	—
8	#11718222	C₄	1	2	yes	—
9	#11718221	—	2	1	no	—
10	#11718214	—	2	1	no	—
11	#11718214	C₄	1	2	yes	complemented
12	C₁₃×#368	C₄	1	1	yes	characteristic, complemented
13	#11718098	C₄	1	1	yes	characteristic, complemented
14	#11718085	—	2	1	no	—
15	#88237	—	4	1	no	—
16	M₁₆×D₁₃	—	4	1	no	—
17	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
18	#88102	—	2	2	no	—
19	#88101	—	4	1	no	—
20	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₁₃×#108	—	2	1	no	—
22	#88062	—	4	1	no	—
23	#88059	—	2	1	no	—
24	C₂×#6090	D₄	1	1	yes	characteristic
25	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
27	C₁₆×D₁₃	D₄	1	1	yes	characteristic
28	C₁₆×D₁₃	Q₈	1	1	yes	characteristic
29	#88036	—	2	2	no	—
30	#7480	—	13	1	no	maximal, Sylow
31	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
32	C₂²⋊#261	—	8	1	no	—
33	C₄⋊#261	—	4	2	no	—
34	C₄×#261	—	8	1	no	—
35	C₁₃×M₁₆	—	4	1	no	—
36	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
37	#6099	—	4	1	no	—
38	C₂×#1055	SD₁₆	1	1	yes	characteristic
39	#6094	—	8	1	no	—
40	C₈×D₁₃	—	8	1	no	—
41	C₈×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₈×D₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₈×D₁₃	SD₁₆	1	1	yes	characteristic
44	C₈×D₁₃	Q₁₆	1	1	yes	characteristic
45	C₁₃×C₁₆	C₄²	1	1	yes	characteristic
46	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
47	#6090	C₂²⋊C₄	1	1	yes	characteristic
48	#6090	C₄⋊C₄	1	1	yes	characteristic
49	C₂×M₆₄	—	13	1	no	—
50	C₂×#368	—	13	1	no	—
51	#2158	—	13	1	no	—
52	C₂²×D₁₃	SD₃₂	1	1	yes	characteristic
53	C₂×#261	—	8	2	no	—
54	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
55	C₂×Dic₁₃	Q₃₂	1	1	yes	characteristic
56	C₄×D₁₃	#102	1	1	yes	characteristic
57	C₄×D₁₃	#107	1	1	yes	characteristic
58	C₄×D₁₃	D₁₆	1	1	yes	characteristic
59	C₄×D₁₃	SD₃₂	1	1	yes	characteristic
60	C₈×C₁₃	—	8	1	no	—
61	C₈×C₁₃	#95	1	1	yes	characteristic
62	C₈×C₁₃	#103	1	1	yes	characteristic
63	#1055	—	8	1	no	—
64	#1055	#103	1	1	yes	characteristic
65	#1055	#106	1	1	yes	characteristic
66	C₂²×C₁₆	—	13	1	no	—
67	C₂×#108	—	26	1	no	—
68	#427	—	26	1	no	—
69	M₆₄	—	13	2	no	—
70	M₆₄	—	26	1	no	—
71	C₂×C₃₂	—	26	1	no	—
72	#368	—	13	1	no	—
73	#368	—	26	1	no	—
74	#368	#261	1	1	yes	characteristic, complemented
75	#367	—	13	2	no	—
76	#366	—	26	1	no	—
77	C₂²×C₁₃	#358	1	1	yes	characteristic
78	C₂×D₁₃	—	4	1	no	—
79	C₂×D₁₃	#357	1	1	yes	characteristic
80	C₂×D₁₃	#367	1	1	yes	characteristic
81	#261	—	16	2	no	—
82	C₄×C₁₃	#340	1	1	yes	characteristic
83	C₄×C₁₃	#357	1	1	yes	characteristic
84	Dic₁₃	#358	1	1	yes	characteristic
85	Dic₁₃	#366	1	1	yes	characteristic
86	C₂×M₁₆	—	52	1	no	—
87	C₂²×C₈	—	13	1	no	—
88	#117	—	52	1	no	—
89	C₂×C₁₆	—	13	1	no	—
90	C₂×C₁₆	—	13	1	no	—
91	C₂×C₁₆	—	13	1	no	—
92	C₂×C₁₆	—	13	1	no	—
93	C₂×C₁₆	—	13	1	no	—
94	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
95	#108	—	2	1	no	—
96	#108	—	26	1	no	—
97	#108	—	52	1	no	—
98	#107	—	26	2	no	—
99	#106	—	52	1	no	—
100	C₃₂	—	26	2	no	—
101	C₂×C₁₃	—	2	1	no	—
102	C₂×C₁₃	#1380	1	1	yes	characteristic
103	D₁₃	—	2	1	no	—
104	D₁₃	—	2	1	no	—
105	C₂²×C₄	—	13	1	no	—
106	M₁₆	—	4	1	no	—
107	M₁₆	—	52	1	no	—
108	M₁₆	—	104	1	no	—
109	C₂×C₈	—	13	1	no	—
110	C₂×C₈	—	13	1	no	—
111	C₂×C₈	—	13	1	no	—
112	C₂×C₈	—	13	1	no	—
113	C₂×C₈	—	13	1	no	—
114	C₂×C₈	—	104	1	no	—
115	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
116	C₄⋊C₄	—	52	2	no	—
117	C₂²⋊C₄	—	104	1	no	—
118	C₄²	—	104	1	no	—
119	C₁₆	—	13	1	no	—
120	C₁₆	—	13	1	no	—
121	C₁₆	C₄×#261	1	1	yes	characteristic
122	C₁₆	C₄⋊#261	1	1	yes	characteristic
123	C₁₃	#7480	1	1	yes	characteristic, complemented, Sylow
124	C₂³	—	13	1	no	—
125	C₂×C₄	—	13	1	no	—
126	C₂×C₄	—	13	1	no	—
127	C₂×C₄	—	13	1	no	—
128	C₂×C₄	—	13	1	no	—
129	C₂×C₄	—	13	1	no	—
130	C₂×C₄	—	104	2	no	—
131	C₂×C₄	#88115	1	1	yes	characteristic
132	C₈	—	8	1	no	—
133	C₈	—	13	1	no	—
134	C₈	—	13	1	no	—
135	C₈	—	104	1	no	—
136	C₈	(C₂×#261)⋊C₄	1	1	yes	characteristic
137	C₈	#88117	1	1	yes	characteristic
138	C₂²	—	13	1	no	—
139	C₂²	—	13	1	no	—
140	C₂²	—	52	1	no	—
141	C₂²	#11718280	1	1	yes	characteristic
142	C₄	—	13	1	no	—
143	C₄	—	13	1	no	—
144	C₄	—	208	2	no	—
145	C₄	#11718246	1	1	yes	characteristic
146	C₄	#11718276	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	26	1	no	—
149	C₂	—	26	1	no	—
150	C₂	#13129893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718214

C₂

yes

maximal, characteristic

#13131187

C₂

yes

maximal, characteristic

D₁₃×#368

C₂

yes

maximal, characteristic

#11719092

—

C₂×C₁₆×D₁₃

C₂²

yes

characteristic

D₁₃×#108

—

C₂×#88036

—

#11718222

C₄

yes

—

#11718221

—

#11718214

—

#11718214

C₄

yes

complemented

C₁₃×#368

C₄

yes

characteristic, complemented

#11718098

C₄

yes

characteristic, complemented

#11718085

—

#88237

—

M₁₆×D₁₃

—

C₂×C₈×D₁₃

D₄

yes

characteristic

#88102

—

#88101

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₁₃×#108

—

#88062

—

#88059

—

C₂×#6090

D₄

yes

characteristic

C₁₆×D₁₃

C₂×C₄

yes

characteristic

C₁₆×D₁₃

C₂×C₄

yes

characteristic

C₁₆×D₁₃

D₄

yes

characteristic

C₁₆×D₁₃

Q₈

yes

—

—

C₂×C₄×D₁₃

D₈

yes

characteristic

C₂²⋊#261

—

C₄⋊#261

—

C₄×#261

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

#6099

—

C₂×#1055

SD₁₆

yes

characteristic

#6094

—

C₈×D₁₃

—

C₈×D₁₃

C₂²⋊C₄

yes

characteristic

C₈×D₁₃

C₄⋊C₄

yes

characteristic

C₈×D₁₃

SD₁₆

yes

characteristic

C₈×D₁₃

Q₁₆

yes

characteristic

C₁₃×C₁₆

C₄²

yes

characteristic

C₁₃×C₁₆

C₄⋊C₄

yes

characteristic

#6090

C₂²⋊C₄

yes

characteristic

#6090

C₄⋊C₄

yes

characteristic

C₂×M₆₄

—

C₂×#368

—

#2158

—

C₂²×D₁₃

SD₃₂

yes

characteristic

C₂×#261

—

C₂×C₄×C₁₃

#102

yes

characteristic

C₂×Dic₁₃

Q₃₂

yes

C₄×D₁₃

yes

C₄×D₁₃

yes

C₄×D₁₃

D₁₆

yes

characteristic

C₄×D₁₃

SD₃₂

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

—

yes

yes

C₂²×C₁₆

—

C₂×#108

—

#427

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

—

—

yes

characteristic, complemented

#367

—

#366

—

C₂²×C₁₃

#358

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

yes

C₂×D₁₃

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

yes

Dic₁₃

yes

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

—

—

—

—

—

100

C₃₂

—

101

C₂×C₁₃

—

102

C₂×C₁₃

#1380

yes

characteristic

103

D₁₃

—

104

D₁₃

—

105

C₂²×C₄

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

104

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

104

—

115

C₂×C₈

C₂²⋊#261

yes

characteristic

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

104

—

118

C₄²

—

104

—

119

C₁₆

—

120

C₁₆

—

121

C₁₆

C₄×#261

yes

characteristic

122

C₁₆

C₄⋊#261

yes

characteristic

123

C₁₃

#7480

yes

characteristic, complemented, Sylow

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

104

—

131

C₂×C₄

#88115

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

104

—

136

C₈

(C₂×#261)⋊C₄

yes

characteristic

137

C₈

#88117

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

#11718280

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

208

—

145

C₄

#11718246

yes

characteristic

146

C₄

#11718276

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

#13129893

yes

characteristic

	subgroup	G/H
center	C₄	#11718276
commutator	C₁₃×C₁₆	C₄²
Frattini	C₂×C₁₆	C₂×#261
Fitting	C₁₃×#368	C₄
socle	C₂×C₁₃	#1380
prime radical	C₂²×D₁₃	SD₃₂
cosocle	C₂×C₁₆×D₁₃	C₂²
hypercenter	#368	#261
hypocenter	C₁₃	#7480
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

subgroup

G/H

center

C₄

#11718276

commutator

C₁₃×C₁₆

C₄²

Frattini

C₂×C₁₆

C₂×#261

Fitting

C₁₃×#368

C₄

socle

C₂×C₁₃

#1380

prime radical

C₂²×D₁₃

SD₃₂

cosocle

C₂×C₁₆×D₁₃

C₂²

C₁₃

G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

N	H ≅ G/N	type
11. #11718214	C₄	semidirect
12. C₁₃×#368	C₄	semidirect
13. #11718098	C₄	semidirect
74. #368	#261	semidirect
123. C₁₃	#7480	semidirect

H ≅ G/N

type

11. #11718214

C₄

semidirect

12. C₁₃×#368

C₄

semidirect

13. #11718098

C₄

semidirect

74. #368

#261

semidirect

123. C₁₃

#7480

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11718214	2
#13131187	2
D₁₃×#368	2
#7480	13

subgroup

index

C₂×#11718214

#13131187

D₁₃×#368

#7480

Subgroups — #16778028

Conjugacy classes of subgroups

Complements

Maximal subgroups